Ring of Flows of $k$-dimensional Autonomous Dynamical Systems

TL;DR

This paper introduces a novel framework called the ring of flows that decomposes complex autonomous nonlinear dynamical systems into manageable parts, enabling exact solutions through a modular approach.

Contribution

It presents a new mathematical structure for decomposing and solving autonomous nonlinear dynamical systems exactly, which was not previously available.

Findings

Allows decomposition of systems into smaller parts

Enables exact solutions through modular assembly

Provides a new tool for analyzing nonlinear dynamics

Abstract

We construct a ring of flows where we can decompose autonomous nonlinear dynamical systems into smaller parts, then solve each part and finally put everything together to obtain the exact solution of these systems.